Quantum point contacts, full counting statistics and the geometry of planes

The description of the full counting statistics (FCS) of charge transport in mesoscopic systems is quite involved even for the zero temperature case. It requires either implementation of the Keldysh technique, or the direct computation of the determinant of an infinite matrix, or solving an auxiliary Riemann- Hilbert problem. Here we present a simple geometrical description of the FCS of charge transport for zero temperature. We show that, the FCS is solely determined by the geometry of two planes. We consider a quantum point contact between two ideal single-channel conductors coupled by a tunnelling barrier and treat the effect of a voltage applied between the contacts. In the particular case of N quantized Lorentzian pulses the computation of the FCS reduces to the diagonalization of an N × N matrix. We also show how our geometrical formulation enables us to compute the overlap between the initial ground state and the final one in the orthogonality catastrophe problem.